Use nuos technology to acquire optimized 2D data

ABSTRACT

A method for 2D seismic data acquisition includes determining source-point seismic survey positions for a combined deep profile seismic data acquisition with a shallow profile seismic data acquisition wherein the source-point positions are based on non-uniform optimal sampling. A seismic data set is acquired with a first set of air-guns optimized for a deep-data seismic profile and the data set is acquired with a second set of air-guns optimized for a shallow-data seismic profile. The data are de-blended to obtain a deep 2D seismic dataset and a shallow 2D seismic dataset.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a non-provisional application which claims benefitunder 35 USC § 119(e) to U.S. Provisional Application Ser. No.62/416,571 filed Nov. 2, 2016, entitled, “USE NUOS TECHNOLOGY TO ACQUIREOPTIMIZED 2D DATA”, which is incorporated herein in its entirety.

FIELD OF THE INVENTION

The present invention relates generally to seismic imaging. Moreparticularly, but not by way of limitation, embodiments of the presentinvention include tools and methods for acquiring and processing seismicdata using compressive sensing with optimized source and sensorsampling.

BACKGROUND OF THE INVENTION

Seismic imaging typically involves not only acquiring seismic data butprocessing the acquired seismic data. In some cases, processing requiresrecovering missing pieces of information from irregularly acquiredseismic data. Irregularities may be caused by, for example, dead orseverely corrupted seismic traces, surface obstacles, acquisitionapertures, economic limits, and the like. Certain seismic processingtechniques may be employed to spatially transform irregularly acquiredseismic data to regularly sampled data that is easier to interpret. Thisregularization can involve a number of processing techniques such asinterpolation and reconstruction of seismic data.

In recent years, compressive sensing theories have gained traction. Oneapplication of compressive sensing in geophysics involves seismic datareconstruction (e.g., Hennenfent and Herrmann, 2008). As an overview,compressive sensing provides conditions for when an under-determinedsystem of equations has a desirable solution. A seismic datareconstruction problem (e.g. Stolt, 2002; Trad, 2003; Liu and Sacchi,2004; Abma and Kabir, 2006; Ramirez et al., 2006; Naghizadeh and Sacchi,2007; Xu et al., 2010; Kaplan et al., 2010) provides a coarse set ofobserved traces along with a desired set of fine spatial grid pointsupon which data is reconstructed. Compressive sensing theory can addresssuch issues as 1) how many observations need to be collected, 2) wherethe observations should be made (i.e., sampling grid) with respect tothe reconstruction grid, and 3) what mathematical dictionary (e.g.,mutual coherence) should be used to represent the reconstructed data.While mutual coherence is an important metric in compressive sensingtheory, it can also be expensive to compute. Descriptions and/oroverviews of seismic data reconstruction can also be found in Trad,2003; Liu and Sacchi, 2004; Abma and Kabir, 2006; Naghizadeh and Sacchi,2007; Xu et al., 2010, which are hereby incorporated by reference,hereafter.

Certain data reconstruction techniques have been developed, whichprovide a sparse representation of reconstructed data. For example, Liuand Sacchi (2004) promote a sparse solution in wave-number domain usinga penalty function constructed from inverse power spectrum of thereconstructed data. In compressive sensing, it is common to apply an

₁ norm to promote some sparse representation of the reconstructed data.The

₁ norm has become of particular interest due to its relation to the

₀ norm which is a count of the number of non-zero elements. Theoremsprovide conditions for exact recovery of the reconstructed data andwhich, in part, rely on relationship between the

₁ and

₀ norms, and use of the

₁ norm in a convex optimization model (Candes et al., 2006). At leastone theory of compressive sensing indicates that a sparse orcompressible signal can be recovered from a small number of randomlinear measurements by solving a convex

₁ optimization problem (e.g. Baraniuk, 2007).

Compressive sensing can also provide new opportunities for survey designusing an irregular sampling grid (e.g. Hennenfent and Herrmann, 2008;Kaplan et al., 2012) instead of a traditional regular grid in order toincrease bandwidth and reduce cost. Generally, irregular survey designbased on compressive sensing can be summarized by the followingsteps: 1) determine a nominal regular grid for survey area, 2) choose asubset of locations from this nominal grid in a random or randomlyjittered (Hennenfent and Herrmann, 2008) fashion, 3) acquire seismicdata based on chosen locations, and 4) reconstruct the data back to theoriginal nominal grid. This approach is applicable to both shot andreceiver dimensions.

In certain cases, compressive sensing using irregular acquisition gridscan be used to recover significantly broader spatial bandwidth thancould be obtained using a regular sampling grid. Recovered bandwidth isprimarily determined according to the spacing of nominal grid forreconstruction. If a predefined nominal grid is too coarse, thereconstructed seismic data may still be aliased; if the predefinednominal grid is too fine, the time and cost savings of irregular versusregular survey design may become insignificant. In general, if there isa lack of prior information about a given survey area, then it may notbe feasible to select a proper nominal grid beforehand.

BRIEF SUMMARY OF THE DISCLOSURE

The present invention relates generally to seismic imaging. Moreparticularly, but not by way of limitation, embodiments of the presentinvention include tools and methods for processing seismic data bycompressive sensing and non-uniform optimal sampling.

Compressive sensing theory is utilized for seismic data reconstruction.Compressive sensing, in part, requires an optimization model. Twoclasses of optimization models, synthesis- and analysis-basedoptimization models, are considered. For the analysis-based optimizationmodel, a novel optimization algorithm (SeisADM) is presented. SeisADMadapts the alternating direction method with a variable-splittingtechnique, taking advantage of the structure intrinsic to the seismicdata reconstruction problem to help give an efficient and robustalgorithm. SeisADM is demonstrated to solve a seismic datareconstruction problem for both synthetic and real data examples. Inboth cases, the SeisADM results are compared to those obtained fromusing a synthesis-based optimization model. Spectral Projected GradientL1 solver (SPGL1) method can be used to compute the synthesis-basedresults. Through both examples, it is observed that data reconstructionresults based on the analysis-based optimization model are generallymore accurate than the results based on the synthesis-based optimizationmodel. In addition, for seismic data reconstruction, the SeisADM methodrequires less computation time than the SPGL1 method.

Compressive sensing can be successfully applied to seismic datareconstruction to provide a powerful tool that reduces the acquisitioncost, and allows for the exploration of new seismic acquisition designs.Most seismic data reconstruction methods require a predefined nominalgrid for reconstruction, and the seismic survey must containobservations that fall on the corresponding nominal grid points.However, the optimal nominal grid depends on many factors, such asbandwidth of the seismic data, geology of the survey area, and noiselevel of the acquired data. It is understandably difficult to design anoptimal nominal grid when sufficient prior information is not available.In addition, it may be that the acquired data contain positioning errorswith respect to the planned nominal grid. An interpolated compressivesensing method is presented which is capable of reconstructing theobserved data on an irregular grid to any specified nominal grid,provided that the principles of compressive sensing are satisfied. Theinterpolated compressive sensing method provides an improved datareconstruction compared to results obtained from some conventionalcompressive sensing methods.

Compressive sensing is utilized for seismic data reconstruction andacquisition design. Compressive sensing theory provides conditions forwhen seismic data reconstruction can be expected to be successful.Namely, that the cardinality of reconstructed data is small under some,possibly over-complete, dictionary; that the number of observed tracesare sufficient; and that the locations of the observed traces relativeto that of the reconstructed traces (i.e. the sampling grid) aresuitably chosen. If the number of observed traces and the choice ofdictionary are fixed, then choosing an optimal sampling grid increasesthe chance of a successful data reconstruction. To that end, a mutualcoherence proxy is considered which is used to measure how optimal asampling grid is. In general, the computation of mutual coherence isprohibitively expensive, but one can take advantage of thecharacteristics of the seismic data reconstruction problem so that it iscomputed efficiently. The derived result is exact when the dictionary isthe discrete Fourier transform matrix, but otherwise the result is aproxy for mutual coherence. The mutual coherence proxy in a randomizedgreedy optimization algorithm is used to find an optimal sampling grid,and show results that validate the use of the proxy using both syntheticand real data examples.

One example of a computer-implemented method for determining optimalsampling grid during seismic data reconstruction includes: a)constructing an optimization model, via a computing processor, given bymin_(u)∥Su∥₁ s.t.∥Ru−b∥₂≤σ wherein S is a discrete transform matrix, bis seismic data on an observed grid, u is seismic data on areconstruction grid, and matrix R is a sampling operator; b) definingmutual coherence as

${\mu \leq \sqrt{\frac{C}{S}\frac{m}{\left( {\log\mspace{14mu} n} \right)^{6}}}},$wherein C is a constant, S is a cardinality of Su, m is proportional tonumber of seismic traces on the observed grid, and n is proportional tonumber of seismic traces on the reconstruction grid; c) deriving amutual coherence proxy, wherein the mutual coherence proxy is a proxyfor mutual coherence when S is over-complete and wherein the mutualcoherence proxy is exactly the mutual coherence when S is a Fouriertransform; and d) determining a sample grid r_(*)=arg min_(r) μ(r).

In one nonlimiting embodiment a method for 2D seismic data acquisitionincludes determining source-point seismic survey positions for acombined deep profile seismic data acquisition with a shallow profileseismic data acquisition wherein the source-point positions are based onnon-uniform optimal sampling. A seismic data set is acquired with afirst set of air-guns optimized for a deep-data seismic profile and thedata set is acquired with a second set of air-guns optimized for ashallow-data seismic profile. The data are de-blended to obtain a deep2D seismic dataset and a shallow 2D seismic dataset.

The method may further comprise using interpolated compressive sensingto reconstruct the acquired dataset to a nominal grid. Additionally, themethod may provide source-point positions based on non-uniform optimalsampling acquired using a Monte Carlo Optimization scheme to determinesource-point seismic survey positions.

The method of using a Monte Carlo Optimization scheme may furthercomprise a Signal-to-Noise Ratio cost-function (SNR cost-function)defined as the root-mean-square SNR of the data to be reconstructedminus the SNR of an elastic wave synthetic dataset over an area ofinterest using an appropriate velocity model.

The method of using a Monte Carlo Optimization scheme may furtherdetermine the non-uniform optimal sampling using a Monte CarloOptimization scheme comprises a cost-function to determine the optimizedlocations, the cost function selected from the group consisting of: i)diagonal dominance, ii) a conventional array response, iii) a conditionnumber, iv) eigenvalue determination, v) mutual coherence, vi) tracefold, or vii) azimuth distribution. In other embodiments, the nominalgrid may be a uniformly sampled grid. The method may further comprisereconstructing the acquired data to obtain one or more receiver gathers.

In another nonlimiting embodiment, the first set of air-guns has a firstencoded source signature and the second set of air-guns has a secondencoded source signature.

In still another nonlimiting embodiment, determining the optimizedsource-point positions is based on non-uniform optimal sampling whichfurther comprises determining an underlying uniformly sampled grid.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete understanding of the present invention and benefitsthereof may be acquired by referring to the follow description taken inconjunction with the accompanying drawings in which:

FIG. 1 illustrates an algorithm as described in Example 1.

FIG. 2 illustrates a plot as described in Example 1.

FIGS. 3a-3d illustrate plots as described in Example 1.

FIGS. 4a-4c illustrate plots as described in Example 1.

FIGS. 5a-5c illustrate Fourier spectra as described in Example 1.

FIG. 6 illustrates a plot as described in Example 2.

FIGS. 7a-7e illustrate plots as described in Example 2.

FIG. 8 illustrates a plot as described in Example 2.

FIGS. 9a-9c illustrate plots as described in Example 2.

FIGS. 10a-10c illustrate plots as described in Example 2.

FIGS. 11a-11b illustrate plots as described in Example 3.

FIGS. 12a-12b illustrate plots as described in Example 3.

FIGS. 13a-13d illustrate plots as described in Example 3.

FIGS. 14a-14d illustrate plots as described in Example 3.

FIG. 15 illustrates a flow chart as described in Example 4.

FIG. 16 illustrates a schematic diagram of an embodiment of a systemaccording to various embodiments of the present disclosure.

DETAILED DESCRIPTION

Reference will now be made in detail to embodiments of the invention,one or more examples of which are illustrated in the accompanyingdrawings. Each example is provided by way of explanation of theinvention, not as a limitation of the invention. It will be apparent tothose skilled in the art that various modifications and variations canbe made in the present invention without departing from the scope orspirit of the invention. For instance, features illustrated or describedas part of one embodiment can be used on another embodiment to yield astill further embodiment. Thus, it is intended that the presentinvention cover such modifications and variations that come within thescope of the invention.

Some embodiments of the present invention provide tools and methods forreconstructing seismic data utilizing compressive sensing. Convexoptimization models used for reconstructing seismic data can fall underat least two categories: synthesis-based convex optimization model andanalysis-based convex optimization model (Candes et al., 2008). As usedherein, the term “convex optimization problem” and its related termssuch as “convex optimization model” generally refer to a mathematicalprogramming problem of finding solutions when confronted withconflicting requirements (i.e., optimizing convex functions over convexsets).

Some embodiments of the present invention provides tools and methods foroptimizing the analysis-based convex optimization model. At least oneembodiment adapts an alternating direction method (Yang and Zhang, 2011)with a variable-splitting technique (Wang et al., 2008; Li, 2011). Thisallows a user to take advantage of the structure in the seismic datareconstruction problem to provide a more efficient solution. Otheradvantages will be apparent from the disclosure herein.

According to one or more embodiments of the present invention, atwo-dimensional windowed Fourier transform representation of the data(e.g. Mallat, 2009) may be provided. In some embodiments, an irregularacquisition grid may be provided, which is an additional condition forexact recovery given by compressive sensing theory. The irregularity inseismic data can be quantified by mutual coherence which is a functionof the irregular acquisition grid and windowed Fourier transform basis(e.g. Elad et al., 2007).

Some embodiments provide tools and methods for interpolated compressivesensing data reconstruction for recovering seismic data to a regularnominal grid that is independent of the observed trace locations.Advantages include, but are not limited to, 1) one can try distinctnominal grids for data reconstruction after acquisition, and 2)positioning errors occurring during acquisition can be taken intoaccount. Other geophysical methods for seismic data reconstruction canrely on the discrete Fourier transform to allow for the arbitraryrelation between observed trace locations and the nominal grid. Bycontrast, in the present invention, the transform (Fourier or otherwise)is applied to the nominal grid, and the burden of the mismatch betweenobserved trace locations and the nominal grid is shifted to arestriction/sampling operator.

Some embodiments provide tools and methods that derive a mutualcoherence proxy applicable to the seismic data reconstruction problem.At least one advantage is that this proxy is efficient to compute. Moreparticularly, it is the maximum non-d.c. component of the Fouriertransform of the sampling grid. A greedy optimization algorithm (e.g.Tropp, 2004) is used to find an optimal sampling grid, with the mutualcoherence proxy giving data independent measure for optimal. Theoptimization problem is typically non-convex, and so the greedyalgorithm finds a locally optimal solution that depends on how thealgorithm is initialized.

Example 1

Data Reconstruction Model

For data reconstruction, a system is defined, wherein (Herrmann, 2010),b=RS*x, x=Su,  (1)where b is observed seismic data, and u is reconstructed seismic data.Matrix R is a restriction (i.e. sampling) operator, mapping from thereconstructed seismic data to the observed seismic data. If S is anappropriately chosen dictionary, then x is a sparse representation of u.For most over-complete dictionaries, such as a wavelet, curvelet andwindowed Fourier transforms,S*S=I  (2)Optimization Models

Given the over-complete linear system in equation 1, and observed datab, solution(s) to the reconstructed data u are computed. A frequentlyused approach from compressive sensing is to solve either basis pursuit(BP) optimization model for noise-free observed data,min_(x) ∥x∥ ₁ s.t. RS*x=b  (3)or the basis pursuit de-noising (BPDN) optimization model for noisy orimperfect observed data,min_(x) ∥x∥ ₁ s.t.∥RS*x−b∥ ₂ ²≤σ  (4)where σ is a representative of the noise level in the observed data. Forexample, if {tilde over (x)} is the solution to the optimization modelin equation 3, thenũ=S*{tilde over (x)}  (5)are reconstructed data. In solving either the BP or BPDN model, anassumption may be made that the reconstructed data u have a sparserepresentation under the dictionary S. Solving the optimization modelsin equations 3 and 4 is often referred to as synthesis-based

₁ recovery (Candes et al., 2008). SPGL1, as proposed by van den Berg andFriendlander (2008), and based on an analysis of the Pareto curve, isone of the most efficient of these methods.

An alternative to the synthesis-based optimization models areanalysis-based optimization models for both the noise-free case,min_(u) ∥Su∥ ₁ s.t. Ru=b  (6)and the noisy case,min_(u) ∥Su∥ ₁ s.t.∥Ru−b∥ ₂≤σ  (7)Solving the optimization models in equations 6 and 7 is calledanalysis-based

₁ recovery (Candes et al., 2008). When the dictionary S is orthonormal,synthesis- and analysis-based models are theoretically equivalent.However, according to Candes et al. (2008), when S is overcompleteanalysis based optimization models involve fewer unknowns and arecomputationally easier to solve than their synthesis-basedcounter-parts. Additionally, analysis-based reconstruction may give moreaccurate solutions than those obtained from synthesis-basedreconstruction (Elad et al., 2007).Alternating Direction Algorithm with Variable Splitting

The SeisADM algorithm performs analysis-based

₁ recovery based on the optimization model in equation 7. SeisADM isbased on the alternating direction method (e.g. Gabay and Mercier, 1976;Glowinski, 1984; Yang and Zhang, 2011). The alternating direction method(ADM) has been widely used to solve inverse problems. It is known as arobust and stable iterative algorithm, but is usually very costly due toits estimation of the gradient for each iteration. Here, a variablesplitting technique in combination with ADM is introduced, whichutilizes the structure of the seismic data reconstruction model to findan efficient method for solving the optimization model in equation 7. Inparticular, the fact that S*S=I, and that R*R is a diagonal matrix areutilized. A similar algorithm can be derived for the noise-free case(equation 6) as well.

Starting from equation 7, splitting variables w=Su is introduced toseparate the operator S from the non-differentiable

₁ norm, and v=Ru−b to form a

₂-ball constrained optimization problem (we only need to introduce onesplitting variable w to solve the noise-free model (equation 6)).Therefore, equation 7 is equivalent to,min_(u,w,v) ∥w∥ ₁ s.t. w=Su, v+b=Ru,∥v∥ ₂≤σ  (8)Ignoring the

₂-ball constraint (∥v∥₂≤σ), equation 8 has the corresponding augmentedLagrangian function (Gabay and Mercier, 1976),

$\begin{matrix}{{L_{A}\left( {w,u,v} \right)} = {{w}_{1} - {\gamma^{*}\left( {{Su} - w} \right)} + {\frac{\beta}{2}{{{Su} - w}}_{2}^{2}} - {\lambda^{*}\left( {{Ru} - b - v} \right)} + {\frac{\mu}{2}{{{Ru} - b - v}}_{2}^{2}}}} & (9)\end{matrix}$where γ and λ are Lagrange multipliers, and β and μ are scalars. SeisADMfinds the minimum of the equivalent model in equation 8. It does so byminimizing the augmented Lagrangian function in equation 9 with respectto, separately, w, u and v, and then updating the Lagrange multipliers,γ and μ.

For constant u and v, the w-subproblem is,

$\begin{matrix}{\min_{w}\left\{ {{w}_{1} - {\gamma^{*}\left( {{Su} - w} \right)} + {\frac{\beta}{2}{{{Su} - w}}_{2}^{2}}} \right\}} & (10)\end{matrix}$Equation 10 is separable with respect to each w_(i)∈w and has theclosed-form solution (e.g. Li, 2011),

$\begin{matrix}{\overset{\sim}{w} = {\max\left\{ {{{{{Su} - \frac{\gamma}{\beta}}}_{1} - \frac{1}{\beta}},0} \right\}{{sgn}\left( {{Su} - \frac{\gamma}{\beta}} \right)}}} & (11)\end{matrix}$where sgn (x) is 1 for x>0, 0 for x=0, and −1 for x<0.

For constant w and v, the u-subproblem is,

$\begin{matrix}{\min_{u}\left\{ {{- {\gamma^{*}\left( {{Su} - w} \right)}} + {\frac{\beta}{2}{{{Su} - w}}_{2}^{2}} - {\lambda^{*}\left( {{Ru} - b - v} \right)} + {\frac{\mu}{2}{{{Ru} - b - v}}_{2}^{2}}} \right\}} & (12)\end{matrix}$Equation 12 is quadratic, with the corresponding normal equations,(βS*S+μR*R)ũ=S*(βw+γ)+R*(μb+μv+λ).  (13)Since S*S=I and R*R is a diagonal matrix, one can explicitly andefficiently solve equation 13.

For constant w and u, the v-subproblem is,

$\begin{matrix}{\min_{v}\left\{ {{{- {\lambda^{*}\left( {{Ru} - b - v} \right)}} + {\frac{\mu}{2}{{{Ru} - b - v}}_{2}^{2}\mspace{14mu}{s.t.\mspace{11mu}{v}_{2}}}} \leq \sigma} \right.} & (14)\end{matrix}$The value of v found from solving equation 14 is equivalent to thatfound from solving,

$\begin{matrix}{{\min_{v}{{{\left( {{Ru} - b - v} \right) - \frac{\lambda}{\mu}}}_{2}^{2}\mspace{14mu}{s.t.\mspace{11mu}{v}_{2}}}} \leq \sigma} & (15)\end{matrix}$

Further, if

$\begin{matrix}{\theta = {{Ru} - b - \frac{\lambda}{\mu}}} & (16)\end{matrix}$then it can be shown that the explicit solution of equation 15 is,

$\begin{matrix}{\overset{\sim}{v} = \left\{ \begin{matrix}{\theta,} & {{{if}\mspace{14mu}{\theta }_{2}} \leq \sigma} \\{{\sigma\frac{\theta}{{\theta }_{2}}},} & {otherwise}\end{matrix} \right.} & (16)\end{matrix}$

The SeisADM algorithm is iterative, where for each iteration γ and λ areheld constant, and the minimum (ũ,{tilde over (v)},{tilde over (w)}) ofthe three sub-problems described above are found. At the end of eachiteration, the Lagrange multipliers (Glowinski, 1984) is updated,

$\begin{matrix}\left\{ \begin{matrix}{\overset{\sim}{\gamma} = {\gamma - {{\xi\beta}\left( {{Su} - w} \right)}}} \\{\overset{\sim}{\lambda} = {\lambda - {{\xi\mu}\left( {{Ru} - b - v} \right)}}}\end{matrix} \right. & (17)\end{matrix}$Provided that

${0 < \xi < \frac{1 + \sqrt{5}}{2}};$the theoretical convergence of ADM can be guaranteed. Putting all thecomponents together, our algorithm for solving the analysis-baseddenoising model (equation 7) is summarized in FIG. 1.Numerical Results

Two tests are performed and reported in this section to demonstrate theanalysis-based

₁ recovery and efficiency of SeisADM. Specifically, SeisADM is comparedwith SPGL1. In an effort to make comparisons fair, an effort can be madeto optimally tune parameters for both SeisADM and SPGL1.

Synthetic Data Example

For a synthetic example, data were generated from the Sigsbee 2a model(Bergsma 2001), and a two-dimensional acoustic finite differencesimulation. In addition, the data were corrupted with random Gaussiannoise, such that the data have a signal to noise ratio of 12.7 dB. Asingle shot gather is reconstructed to where a set of observed receiversare reconstructed to a regular grid with 1300 receivers with 7:62 mbetween adjacent receivers. In running 111 data reconstructionsimulations, for each simulation the size of the set of observed traceschanged, ranging from 8% to 50% of the total number of reconstructedtraces.

The results are shown in FIG. 2 which plots the signal to-noise ratio asa function of the percentage of reconstructed traces that are in theobservation set. Results for both synthesis-based

₁ recovery (using SPGL1), and analysis based

₁ recovery (using SeisADM) are shown. In addition, the horizontal linein FIG. 2 is the signal-to-noise ratio in the original data. Thesignal-to-noise ratio values in FIG. 2 are computed as a function of thereconstructed and noise-free data, and where the noise-free data areproduced from the finite difference simulation.

FIGS. 3a-3d illustrate (for a small window of time and receivers)reconstruction results for when the number of observed traces is 15% ofthe number of the reconstructed traces. In particular, FIG. 3a is thefinite difference simulated data on the reconstruction grid (i.e. thetrue result), FIG. 3b is the set of observed traces, FIG. 3c is thesynthesis-based

₁ reconstruction result, and FIG. 3d is the analysis-based

₁ reconstruction result. Finally, the computation time to produce thesynthesis-based result using SPGL1 was 95 s, while the computation timeto produce the analysis-based result using SeisADM was 64 s.

FIG. 2 shows that when the number of observed traces is less than 20% ofthe number of reconstructed traces, analysis-based

₁ recovery using SeisADM provides significantly higher quality (i.e.higher signal-to-noise ratio) than synthesis-based

₁ recovery using SPGL1. From FIGS. 3a-3d , qualitatively less noise wasobserved, for example, in the central part, in the SeisADM resultcompared to the SPGL1 result.

Real Data Example

For a real data example, data that were collected with a two-dimensionalocean bottom node acquisition geometry was used. The survey was,specifically, designed in such a way that the shots are recorded on anirregular acquisition grid. The observed data are reconstructed to aregular shot grid with 3105 shot points and 6.25 m between adjacentshots. The observed data for reconstruction are comprised of 564 ofthese 3105 shot points, giving a set of observed shots that isapproximately 18% of the reconstructed shot points. The results are fora single ocean bottom node (common receiver gather).

FIG. 4a-4c show for a small window of shot points and time, commonreceiver gathers. In particular, FIG. 4a plots the observed data on the6.25 m grid. FIG. 4b is the reconstruction result using thesynthesis-based optimization model and the SPGL1 method, and FIG. 4c isthe reconstruction result using the analysis-based optimization modeland the SeisADM method. The seismic event at approximately 3.9 s isbelieved to be of higher quality in the analysis-based result (FIG. 4c )as compared to the synthesis-based result. In FIGS. 5a-5c , thecorresponding f-k spectra of the data and reconstructions are plotted.In particular, FIG. 5a is the Fourier spectrum of the data, FIG. 5b isthe Fourier spectrum of the synthesis-based result, and FIG. 5c is theFourier spectrum of the analysis-based result. The run-time for thesynthesis-based result using SPGL1 was 446 s compared to a run-time of1349 s for the analysis-based result using SeisADM. The f-k spectrum ofthe SeisADM result appears to contain less aliased energy than the f-kspectrum of the SPGL1 result.

CONCLUSIONS

In this Example, the seismic data reconstruction problem usingcompressive sensing was considered. In particular, the significance ofthe choice of the optimization model, being either synthesis- oranalysis-based was investigated. The analysis-based

₁ recovery gave more accurate results than synthesis-based

₁ recovery. A new optimization method for analysis-based

₁ recovery, SeisADM was introduced. SeisADM takes advantage of theproperties of the seismic data reconstruction problem to optimize itsefficiency. The SeisADM method (used for analysis-based

₁ recovery) required less computation time and behaved more robust, ascompared to the SPGL1 method (used for synthesis based

₁ recovery). While the application of SeisADM was to the reconstructionof one spatial dimension, this method may be extended tomulti-dimensional data reconstruction problems.

Example 2

First, the grids used in this Example are defined: 1) the observed gridis an irregular grid on which seismic data are acquired (i.e. observedtrace locations), 2) the nominal grid is a regular grid on which seismicdata are reconstructed, and 3) the initial grid is a regular grid fromwhich the observed grid is selected using, for example, a jitteredsampling scheme.

Traditionally, it is assumed that the initial grid is identical to thenominal grid, and the observed grid lies on a random or jittered subsetof the nominal grid. Under these settings, the model from Herrmann andHennenfent (2008) may be utilized,b=Ru, x=Su  (18)where b=[b₁, . . . , b_(m)]^(T) are observed or acquired seismic data,and u=[u₁, . . . , u_(n)]^(T) (m<n) are data on the nominal grid (i.e.,the true data). Each of b_(i) and u_(i) represents a seismic trace. Theoperator S is an appropriately chosen dictionary which makes Su sparseor approximately sparse, and R is a restriction/sampling operator whichmaps data from the nominal grid to the observed grid. Specifically, R iscomposed by extracting the corresponding rows from an identity matrix.One can recover u by solving an analysis-based basis pursuit denoisingmodel (Cand'es et al., 2008),min_(u) ∥Su∥ ₁ s.t.∥Ru−b∥ ₂≤σ  (19)where s corresponds to the noise level of the observed data. Manyalgorithms have been developed to solve this model or its variants, suchas SPGL1 (van den Berg and Friendlander, 2008), NESTA (Becker et al.,2009), and YALL1 (Yang and Zhang, 2011).Interpolated Compressive Sensing

If the observed grid is independent of the nominal grid, then thenominal grid can be determined after data acquisition. To generalize theidea of compressive sensing seismic data reconstruction, the fact thatseismic data can be well approximated, locally, using a kth-orderpolynomial on a regular grid is utilized. For example, k=1 if theseismic data are linear in a local sense. For the sake of clarity,reconstruction of seismic data is shown along one spatial dimension, butthe method can be easily extended to higher dimensions.

Denoted are the true locations on the observed grid as p₁, . . . , p_(m)and the true locations on the nominal grid as l₁, . . . , l_(n). Forj=1, . . . , m and k<<n,s _(j)

argmin_(s∈{1, . . . ,n−k})Π_(i=0) ^(k) |p _(j) −l _(s+i)|  (20)This is easy to solve due to the fact that l₁, . . . , l_(n) are equallyspaced. When p_(j) is not close to the boundary of the nominal grid,l _(s) _(j) _(+└k/2−1┘) ≤p _(j) ≤l _(s) _(j) _(+└k/2┘+1).  (21)Based on the assumption made at the beginning of this section, givenu_(s) _(j) , . . . , u_(s) _(j) _(+k) for any j=1, . . . , m, b_(j) canbe well approximated using kth-order Lagrange interpolation (e.g. Berrutand Trefethen, 2004); i.e.,

$\begin{matrix}{{b_{j} \simeq {\sum_{i = 0}^{k}{L_{j,{s_{j} + i}}u_{s_{j} + i}}}}{{where},}} & (22) \\{L_{j,{s_{j} + i}} = {\prod\limits_{{h = 0},{h \neq i}}^{k}\left( \frac{p_{j} - l_{s_{j} + h}}{l_{s_{j} + i} - l_{s_{j} + h}} \right)}} & (23)\end{matrix}$

Supposing that u(x) denotes the continuous seismic data in some localwindow, and u(x) is at least k+1 times continuously differentiable.According to the Taylor expansion, the error estimation of Lagrangeinterpolation is

$\begin{matrix}{{e_{j}} = {\frac{{u^{({k + 1})}\left( \xi_{j} \right)}}{\left( {k + 1} \right)!}{\prod\limits_{i = 0}^{k}{{p_{j} - l_{s_{j} + 1}}}}}} & (24)\end{matrix}$for some l_(s) _(j) ≤ξ_(j)≤l_(s) _(j+k) . This also implies the choiceof s_(j) as defined in equation 3.

Inspired by equation 5, interpolated restriction operator is

$\begin{matrix}{{\overset{\sim}{R} = \begin{pmatrix}{{\overset{\sim}{r}}_{1,1}I} & \ldots & {{\overset{\sim}{r}}_{1,n}I} \\\vdots & \ddots & \vdots \\{{\overset{\sim}{r}}_{m,1}I} & \ldots & {{\overset{\sim}{r}}_{m,n}I}\end{pmatrix}}{{where},}} & (25) \\{{\overset{\sim}{r}}_{j,i} = \left\{ \begin{matrix}{L_{j,i},} & {{{{if}\mspace{14mu} s_{j}} \leq i \leq {s_{j} + k}},} \\{0,} & {otherwise}\end{matrix} \right.} & (26)\end{matrix}$and the size of the identity matrix I is decided by the number of timesamples. Then equation 22 can be rewritten as,b≃{tilde over (R)}u  (27)

This demonstrates an embodiment of the interpolated compressive sensingmodel for seismic data reconstruction. Analogous to equation 19, u canbe recovered by solving the following optimization problem,min_(u) ∥Su∥ ₁ s.t.∥{tilde over (R)}u−b∥ ₂≤σ  (28)

One should note that the method described above is fundamentallydifferent from the method which first interpolates the observed databack to nearest points on the nominal grid and then reconstructs usingtraditional compressive sensing. The proposed method utilizes theunknown data on the nominal grid as an interpolation basis to match theobserved data and forms an inverse problem to recover the unknown data.Theoretically, the interpolation error is O(Δh^(k+1)) where Δh is theaverage spacing of the interpolation basis. Since the nominal grid ismuch finer than the observed grid (i.e., smaller average spacing),interpolated compressive sensing is expected to be more accurate thanfirst interpolating followed by reconstructing. Moreover, forinterpolated compressive sensing, the error could be further attenuatedby solving a BP denoising problem such as in equation 28 (Candès et al.,2008).

The computational cost is usually dominated by evaluating {tilde over(R)}^(T){tilde over (R)}u and S^(T)Su at each iteration, which isapproximately O(kN) and O(N log N) respectively, assuming S has a fasttransform (N is the number of samples). Therefore, for seismic datareconstruction, the computational cost for solving the interpolatedcompressive sensing problem in equation 28 is comparable to solving thetraditional compressive sensing problem in equation 19 when k<<N. As theorder k increases, the accuracy of reconstruction may become higher atthe cost of increasing computational burden.

If k=1 in equation 22, then our method is called linear-interpolatedcompressive sensing. Likewise, if k=3, our method is calledcubic-interpolated compressive sensing. In our tests, linear- andcubic-interpolated compressive sensing give comparable and satisfactoryreconstruction results. The case k>3 may only apply to few extremecases. The following data examples focus on the linear- andcubic-interpolated compressive sensing data reconstruction.

Synthetic Data Example

In order to simulate the scenario that the nominal grid does notnecessarily include the observed grid, and also be able to doquantitative analysis, start with a finer initial grid for jitteredsampling, and select a uniform subset from the initial grid as thenominal grid for reconstruction and computing signal-to-noise ratios.The

₁ solver used to solve the problem in equation 28 is based on thealternating direction method proposed by Yang and Zhang (2011).Specifically, the results from two special cases—linear and cubic—of theproposed interpolated compressive sensing with the results fromtraditional compressive sensing are compared. In an effort to make fairnumerical comparisons, the same solver for both traditional andinterpolated compressive sensing is used.

For the synthetic example, data generated from the Sigsbee 2a model(Bergsma, 2001) and a two-dimensional acoustic finite-differencesimulation are considered. For each common receiver gather, the data arereconstructed to a nominal grid with 306 shot points, with a spacing of22.89 m between adjacent shot points. The observed shot points areselected from a regular shot point grid with 7.62 m spacing using ajittered algorithm (Hennenfent and Herrmann, 2008). Experiments wereperformed where the number of observed shot points varies from 15% to50% of the 306 grid points on the nominal grid. There was a mismatchbetween the nominal grid for reconstruction and the initial grid used togenerate the observations; therefore, an observed shot-point does notnecessarily correspond to any given point on the reconstruction grid,making the interpolated compressive sensing method applicable.

The signal-to-noise ratios for reconstruction results is shown in FIG. 6for traditional compressive sensing (with nearest neighbor resampling),linear-interpolated compressive sensing, and cubic-interpolatedcompressive sensing. For reference, data reconstruction results forlinear and cubic interpolation are shown, but without the use ofcompressive sensing data reconstruction. For each data point in FIG. 6,an average signal-to-noise ratio computed from performing 20 datareconstructions, each on a different common receiver gather may be used.FIGS. 7a-7e show data reconstruction results for a small window ofsource points and time of a common receiver gather when there are 108traces in the observed common receiver gather (35% of the reconstructedtraces). In particular, FIG. 7a shows data on the nominal reconstructiongrid, computed using the finite-difference simulation, and FIG. 7b showsthe observed data used for reconstruction. The remaining sections inFIG. 7 show data reconstruction results for traditional compressivesensing (FIG. 7c ), linear-interpolated compressive sensing (FIG. 7d ),and cubic-interpolated compressive sensing (FIG. 7e ).

A qualitative inspection of FIG. 7, confirms the quantitative resultsshown in FIG. 6. Namely that linear- and cubic-interpolated compressivesensing data reconstruction perform similarly, and, for this scenario,provides a large uplift in the signal-to-noise ratio as compared totraditional compressive sensing data reconstruction. In addition, alltypes of compressive sensing data reconstruction outperform datareconstruction using linear and cubic interpolation.

Real Data Example

Marine data was used which were collected by shooting in an irregularacquisition pattern and recorded with a two-dimensional ocean bottomnode acquisition geometry. Two reconstruction experiments using thisdataset were utilized. In the first, the observed data are reconstructedto a nominal shot grid with 2580 shot points and 7.5 m spacing betweenadjacent shots. In the second, the observed data are reconstructed to anominal shot grid with 2037 shot points and 9.5 m spacing betweenadjacent shots. The observed data for reconstruction are comprised of400 shot points that are selected from an initial grid with 6.25 mspacing between adjacent shots points, and 3096 grid points. Similar tothe synthetic example, there is a mismatch between the nominal grids forreconstruction, and the initial grid used to collect the data.Therefore, as before, an observed shot point does not necessarilycorrespond to any given point on the nominal grid.

FIG. 8 shows the acquired data for a small window of time and sourcepoints. Reconstruction results are shown for the 7.5 m nominal grid inFIGS. 9a-9c for the same window of time and source points. Inparticular, traditional (FIG. 9a ), linear-interpolated (FIG. 9b ), andcubic-interpolated (FIG. 9c ) compressive sensing data reconstructionresults are shown. Similarly, the results for the 9.5 m nominal gridwithin the same window of time and source points are shown in FIG.10a-10c , where the traditional, linear-interpolated, andcubic-interpolated compressive sensing data reconstruction results areshown in FIGS. 10a, 10b and 10c respectively.

Even though the seismic data are reconstructed to different nominalgrids with different spacing, the results shown in FIGS. 9 and 10 areconsistent with each other. In both cases, although the effect issubtle, the seismic data recovered using interpolated compressivesensing show less acquisition footprint. Besides, the expression of theseismic events now, for instance, in the lower right hand corner seemsmore geologically plausible than the traditional compressive sensingresult might suggest.

CONCLUSIONS

A novel data reconstruction method, interpolated compressive sensing hasbeen developed. The method allows for a mismatch between the nominalgrid that the data are reconstructed to, and the observed grid uponwhich the data are acquired. This method allows for any dictionary, usedin the compressive sensing data reconstruction model, to be applied tothe regular nominal grid. The relationship between the observed andnominal grids is given by the interpolated restriction operator. Theinterpolated restriction operator, in turn, accounts for both thereduced size of the observed grid, and for when a point on the observedgrid does not correspond to a nominal grid point. The latter is done byincorporating Lagrange interpolation into the restriction operator. Theinterpolated compressive sensing method was applied to both syntheticand real data examples, incorporating both 1st and 3rd order Lagrangeinterpolation into the interpolated restriction operator. The syntheticresults compare linear- and cubic-interpolated compressive sensing totraditional compressive sensing, showing a significant increase in thesignal-to-noise ratio of the reconstructed data. Finally, the method wasapplied to a real data example, and observed an uplift in quality ascompared to traditional compressive sensing.

Example 3

This example finds the optimal sampling grid in a seismic datareconstruction problem. The seismic data reconstruction model can bedescribed as (e.g. Herrmann, 2010),b=Dx, D=RS*, x=Su  (29)where b are seismic data on the observed grid, and u are data on thereconstruction grid (i.e. the true data). The matrix R is a restriction(i.e. sampling) operator, and maps data from the reconstruction grid tothe observed grid. If S is a suitably chosen, possibly over-complete,dictionary, then x will have small cardinality (i.e.

₀-norm).Compressive Sensing Optimization Model and Mutual Coherence

Given the under-determined system in equation 29 and the data b, thereconstructed seismic data u is found by solving an analysis-based basispursuit denoising optimization model (e.g. Candès et al., 2008),min_(u) ∥Su∥ ₁ s.t.∥Ru−b∥ ₂≤σ  (30)

There are many algorithms that can be employed to find the solution ofthe optimization model in equation 30. In this Example, a variant (Li etal., 2012) of the alternating direction method (e.g. Yang and Zhang,2011) is used. At least one goal is to design R (i.e. the sampling grid)such that for a given b and S, u is more likely to be recoveredsuccessfully.

Compressive sensing provides theorems that give conditions for asuccessful data reconstruction. For the moment, we consider thefollowing scenario: 1) S∈R^(n×n) is an orthonormal matrix, 2) R∈R^(m×n)with n>m, 3) D=RS* is such that D is a selection of m rows from S*, and4) D=RS* is such that the columns of D, d_(i), have unit energy∥(d_(i)∥₂=1, i=1 . . . n). Under this scenario, solving the optimizationprogram in equation 30 recovers u successfully with overwhelmingprobability when (Candès et al., 2006),

$\begin{matrix}{\mu \leq \sqrt{\frac{C}{S}\frac{m}{\left( {\log\mspace{11mu} n} \right)^{6}}}} & (31)\end{matrix}$

In equation 31, C is a constant, and S is the cardinality of Su.Importantly, for our analysis, μ is the mutual coherence and is afunction of S and R. In particular (Donoho and Elad, 2002),μ(R,S)=max_(i≠j) |d* _(i) d _(j) |, i,j=1 . . . n  (32)

This is equivalent to the absolute maximum off-diagonal element of theGram matrix, G=D*D. Within the context of the seismic datareconstruction problem, n is proportional to the number of seismictraces on the reconstruction grid, and m is proportional to the numberof traces on the observed grid. Therefore, if S and C are constant, thenfor a given number of observed traces, decreasing m increases the chanceof a successful data reconstruction.

The relation between mutual coherence (equation 32) and the conditionfor exact recovery (equation 31), make its analysis appealing.Unfortunately, for problems in seismic data reconstruction it would beprohibitively expensive to compute. However, if S is the discreteFourier transform matrix, then one can find an efficient method tocompute mutual coherence, and use this as a mutual coherence proxy forwhen S is some over-complete (but perhaps still Fourier deriveddictionary such as the windowed Fourier transform.

To derive the mutual coherence proxy, one may begin by followingHennenfent and Herrmann (2008), and note that for the seismic datareconstruction model, R*R is a diagonal matrix with its diagonal beingthe sampling grid,

$\begin{matrix}{{r = \begin{bmatrix}r_{1} & r_{2} & \ldots & r_{n}\end{bmatrix}}{{hence},}} & (33) \\{\left\lbrack {R^{*}{RS}^{*}} \right\rbrack_{i,j} = {{\sum\limits_{k = 1}^{n}{\left\lbrack {R^{*}R} \right\rbrack_{i,k}\left\lbrack S^{*} \right\rbrack}_{k,j}} = {r_{i}\lbrack S\rbrack}_{i,j}}} & (34)\end{matrix}$and the Gram matrix is,

$\begin{matrix}{\lbrack G\rbrack_{i,j} = {\left\lbrack {D^{*}D} \right\rbrack_{i,j} = {\left\lbrack {{SR}^{*}{RS}^{*}} \right\rbrack_{i,j} = {\sum\limits_{k = 1}^{n}{{\lbrack S\rbrack_{i,k}\left\lbrack S^{*} \right\rbrack}_{k,j}r_{k}}}}}} & (35)\end{matrix}$If S is a discrete Fourier transform matrix, then [S]_(i,j)=ω^(ij) whereω=exp(−2π√{square root over (−1)}/n), and from equation 35,

$\begin{matrix}{\lbrack G\rbrack_{i,j} = {\left\lbrack {D^{*}D} \right\rbrack_{i,j} = {\sum\limits_{k = 1}^{n}{r_{k}\omega^{k{({i - j})}}}}}} & (36)\end{matrix}$Equation 36 shows that off-diagonal elements of the Gram matrix areequal to the non-d.c. components of the Fourier transform of thesampling grid r. Therefore,

$\begin{matrix}{{\mu(r)} = {{\max_{l \neq 0}{{\hat{r}}_{l}}} = {\max_{l \neq 0}{{\sum\limits_{k = 1}^{n}{r_{k}\omega^{kl}}}}}}} & (37)\end{matrix}$where {circumflex over (r)}_(l) are Fourier transform coefficients.Equation 37 can be computed efficiently using the fast Fouriertransform, and is our mutual coherence proxy. It is exactly the mutualcoherence when S is the Fourier transform, and a proxy for mutualcoherence when S is some over-complete dictionary.Greedy Optimization Algorithm for Acquisition Design

Given the mutual coherence in equation 37, a sampling grid r accordingto the optimization program isr _(*) =arg min_(r) μ(r)  (38)where μ is given by equation 37. The optimization program in equation 38is non-convex. To find its solution, a randomized greedy algorithm isproposed. One can think of it as a deterministic alternative to thestatistical result found in Hennenfent and Herrmann (2008). Thealgorithm will find a local minimum, and, therefore, does not guaranteeconvergence to a global minimum. However, in practice, it has beenobserved that solutions finding a local minimum using our randomizedgreedy algorithm are sufficient.

The randomized greedy algorithm for solving equation 38 is shown inAlgorithm 1. The algorithm is initialized using a regular sampling grid,where the spacing of the regular grid is Δr=n/m, so that for any integerj∈{0, 1, . . . , m−1}, the elements of r (equation 33) are,r _(i)={1, i=jΔr+1 0, i≠jΔr+1  (39)and where for the sake of simplicity in our description, one can assumethat n is an integer multiple of m. Dividing the reconstruction gridinto m disjoint subsets of size Δr grid points, and where the jth subsetis,s _(j) ={jΔr−└Δr/2┘+k|k=1 . . . Δr}  (40)where └x┘ denotes integer component of x. In other words, except at theboundaries of the grid, the jth subset is centered on the jth grid pointof the regular observed grid. The ordered sets s_(j) are stored in I,and we store a corresponding random ordering of these sets using J=PI,and where P is a random perturbation matrix. The algorithm sequentiallysteps through the sets in J, and uses a jittering technique so that foreach of the Δr elements in s_(j), its corresponding grid point is set to1 while all others are set to 0, producing a new sampling grid r_(k).Subsequently, the mutual coherence μk=μ (r_(k)) is computed usingequation 37, and compared to the mutual coherence of r. If aperturbation,k _(*) =arg min_(k) μ(r _(k))  (41)on r is found that reduces the mutual coherence, then r is set to r_(k)before iterating to the next sets s_(j)∈J. Hence, the algorithm runs ina fixed number of iterations equal to mΔr, and where the expense at eachiteration is dominated by the computation of the mutual coherence of thesampling grid computed via the fast Fourier transform (equation 37).Therefore, the total expense of the algorithm is O(n² log n).

Algorithm 1 Randomized greedy algorithm r ← 0, Δr ← n / m r_(i) ← 1, fori = jΔr, j = 0,1...,m − 1 s_(j) ← {jΔr − └Δr / 2┘ + k|k = 1...Δr}, j =0,1,...,m − 1 I ← [s₀ s₁ ... s_(m−1)], J ← PI for j = 0 → m − 1  do   s_(j) ← [J]_(j), μ₀ = μ(r), r′ ← r    for ∀k ∈ s_(j)  do       r_({s)_(j) _(})′ ← {0}, r_(k)′ ← 1, μ_(k) ← μ(r′)    end for    If min{ μ_(k)} < μ₀  then       μ₀ ← min {μ_(k) }, r ← r′    end if end for r* ← rSynthetic Data Example

For a synthetic data example, data generated from the Sigsbee 2a model(Bergsma, 2001), and a two-dimensional acoustic finite differencesimulation were used. The data reconstruction of a single commonreceiver gather, and where 184 observed traces are reconstructed to aregular grid with 920 sources and 7.62 m between adjacent sources wereconsidered. Hence, the observed data has 20% as many traces as thereconstructed data. In the data reconstruction model (equation 29), Swas allowed be a two-dimensional windowed Fourier transform.

The results are shown in FIG. 11a-11b which plot the mutual coherenceproxy (equation 38) versus the iteration of the randomized greedyalgorithm. In total there are 185 iterations, including itsinitialization to a regular sampling grid. For comparison, a Monte Carlosimulation generated 185 realizations of R where for each row of R, itsnon-zero column is selected using a random jitter technique with auniform probability density function.

The Monte Carlo realizations of the restriction operator R give,consistently, small values for their mutual coherence proxy (FIG. 11a ),and correspondingly good values for the signal-to-noise ratios of thereconstructed data, as shown in FIG. 11b . This is an expected result,and is shown in Hennenfent and Herrmann (2008). As the greedyoptimization algorithm iterates, the mutual coherence approaches andthen surpasses the mutual coherence computed from the Monte Carlorealizations. Likewise, the signal-to-noise ratios found from therandomized greedy optimization algorithm approach similar values tothose found from the Monte Carlo method. The optimal sampling grid usingthe greedy algorithm was achieved by jittering 134 of the 184observations, a result that is not necessarily predicted by the analyticresult in Hennenfent and Herrmann (2008). However, both the Monte Carloand randomized greedy algorithms produced sampling grids that result insuccessful seismic data reconstructions.

Real Data Example

For the real data example, data that was collected with atwo-dimensional ocean bottom node acquisition geometry were used. Thesurvey was, specifically, designed in such a way that the shots arerecorded on an irregular acquisition grid. The observed data isreconstructed to a regular shot grid with 3105 shot points and 6.25 mbetween adjacent shots. The observed data for reconstruction iscomprised of 400 of these 3105 shot points, giving a set of observedshots that is approximately 13% of the size of the set of reconstructedshot points. The results for a single ocean bottom node (common receivergather) is shown. As was the case for the synthetic data example, S wasallowed be a two-dimensional windowed Fourier transform.

Amplitude spectra of the sampling grids (|{circumflex over (r)}_(l)| inequation 37) are shown in FIGS. 12a-12b . In particular, FIG. 12a showsthe Fourier spectrum of a sampling grid with a large mutual coherenceproxy (μ=176) and FIG. 12b shows the Fourier spectrum of a sampling gridwith a small mutual coherence proxy (μ=99). As expected, the largemutual coherence case corresponds to larger non-d.c. components than thelow mutual coherence case. As shown, the f k-spectra of common receivergathers for the high (FIGS. 13a and 13c ) and low (FIGS. 13b and 13d )mutual coherence sampling grids. In particular, FIGS. 13a-13b plot the fk-spectra of the observed data, and FIGS. 3c-d plot the f k-spectra ofthe reconstructed data. In the f k-spectra of the high mutual coherencecase, coherent aliased energy (FIGS. 13a and 13c ) was observed. In thelow mutual coherence case (FIGS. 13b and 13d ), this energy isattenuated.

Finally, FIGS. 14a-14d plot a common receiver gather for a small windowof time and source points before and after data reconstruction for thelow and high mutual coherence sampling grids. In particular, FIGS.14a-14b plot, respectively, the observed and reconstructed data for thehigh mutual coherence sampling grid, and FIGS. 14c-14d plot,respectively, the observed and reconstructed data for the low mutualcoherence sampling grid. Qualitatively, the acquisition footprint iseasier to see in the high mutual coherence case, as compared to the lowmutual coherence case. These observations are consistent with those madeon the corresponding f k-spectra (FIG. 13).

Example 4

The Non-Uniform Optimal Sampling (NUOS) technology may be used to firetwo different sized gun arrays independently in a 2D sense to acquiredata that is optimized both for the deep geological section and theshallow geological section at the same time. The common problem withdeep array targeting is that it is geared for low frequencies and tendsto be sampled relatively sparsely due to seismic record lengths. Forexample, one embodiment may comprise 37.5 m shot separation on 18 secondrecords. Shallow data on these ‘deep’ records may be degraded andunder-sampled spatially. A dataset directed to shallow data may beacquired at 12.5 m shots and 5 second records. It would be beneficial tobe able to obtain both datasets at the same time or obtain one datasetthat is separable, so that only one pass of the acquisition equipment isrequired in a survey area.

Using NUOS methodology, one embodiment provides for acquisition of thedeep tow gun array on, for example, the port guns and use the starboardgun array to shoot for shallow targeted acquisition, so that the shallowand deep data are simultaneously recording, providing a very efficientacquisition that requires one pass over the survey area instead of two.Each gun array will have its own unique encoding, selected to be asincoherent as practical relative the other array. Then the deep andshallow data may be acquired independently and then the recordsseparated or de-blended after acquisition. Each gun array can fireindependently of the other and not substantially interfere. The methodcuts costs of acquisition in half compared to the conventional approach.

The two different gun arrays are tuned for different objectives in amarine 2D towed streamer survey. Both objectives may be acquiredindependently with a single pass of the vessel and the data de-blendedinto independent 2D lines. Embodiments provided allow acquisition oftwice as much data with optimal sampling for the acquisition costs ofone 2D line instead of the costs of acquiring two lines. The problem oftrying to acquire both deep and shallow data with one gun array or notgetting one or the other dataset is avoided.

A long-standing issue in data acquisition and processing has beenselecting optimal locations for sources and receivers. It is understoodthat random sampling may be able to recover broader bandwidth from afixed set of samples, which may be random, than from uniform sampling.As an improvement over random sampling, embodiments herein providemethodologies for improved means for selecting source and receiverlocations in seismic surveys. NUOS uses concepts from compressivesensing along with optimization algorithms to identify source or sensorpositions that satisfy optimization constraints for a particular survey.After optimizing source or sensor locations, the NUOS approach then usescompressive sensing algorithms to recover significantly broader spatialbandwidth from non-uniform sampling than would be obtained usingconventional uniform sampling. For example, NUOS technology recoversspatial bandwidth equivalent to 12.5 m uniform sampling using the samenumber of samples as would be used for a 25 m sampled survey, withsignificant improvements over conventional surveys. The technology maybe used to reduce costs per area or to survey a larger area with thesame amount of equipment, or to increase survey resolution.

In NUOS source design for one example embodiment, a nominal 37.5 metershot spacing from each gun position may vary between 25 meters to 50meters. This can provide a reconstructed equivalent spacing of 12.5meters. This increases in-line resolution and improves denoising anddemultiple workflows.

According to conventional Nyquist sampling theory, survey layout designwould not be an issue if the earth were sampled to two points perwavelength in each dimension. Practically, orders of magnitude fewersampling points than Nyquist theory would dictate are obtained inconventional or normal surveys. Limited sources and receivers is aclassic “Np-Complete” problem, in that an optimal solution can only befound by investigating every possible combination of source and receiverlocations. Fold and azimuth distribution are examples of cost functionswe routinely use for seismic survey design.

More advanced cost functions for survey design obviously make theoptimization problem more complex, as the computational cost ofevaluating a single solution increases. Compressive sensing (i.e.Baraniuk, 2007) and convex optimization (i.e. Friedlander and Martinez,1994) provide tools to address the seismic survey design problem.Compressive sensing provides for extracting (or ‘reconstructing’) auniformly sampled wave-field from non-uniformly sampled sensors, andconvex optimization provides computationally viable solutions forNp-complete problems. For these methods, a sparse representation of theseismic wave-field must exist in some domain. Algorithms that exploitthe sparsity of seismic wave-fields make the adoption of CompressiveSensing (CS) concepts a natural fit to geophysics.

In recent years, random sampling has been proposed as a means forextracting more bandwidth from seismic data than Nyquist sampling wouldpredict (Herrman, 2010, Moldoveanu, 2010, Milton, et. al., 2012). Use ofnon-uniform sampling for improving signal bandwidth has a long history,having been used in many imaging fields such as signal processing(Shapiro and Silverman, 1960), beamforming (Griffiths and Jim, 1982),synthetic aperture radar (Munson and Sanz, 1984) and seismic imaging(Mosher and Mason, 1985). Non-uniform sampling requires more preciseknowledge of sensor positions than is normally required for uniformsampling. The advent of the Global Positioning System (GPS) incombination with advancements in compressive sensing, optimization, andhigh performance computing makes this NUOS technology practical.

With NUOS, an optimization loop is used to determine the locations ofsources and receivers for a non-uniform design, rather than relyingsolely on decimation, jittering, or randomization. Construction of theoptimization loop requires a cost function that determines the viabilityof a given survey design, and an algorithm for searching the very largespace of possible solutions. The cost function can take many forms,ranging from conventional array response to more sophisticated matrixanalysis techniques (i.e. diagonal dominance, condition number,eigenvalue ranking, mutual coherence, etc). Practical implementations ofNUOS can exploit knowledge of the underlying earth model if known,rather than a model independent compressive sensing implementation.

A cost function is a geophysical attribute of the survey to beoptimized. At the simplest level may be just the fold, (number of hits(traces) in a particular bin) or the offsets (a hit in a particularunique offset plane in a bin) or the azimuth distribution (hits in aparticular bin coming from a particular direction) are all things to beoptimized and keep relatively uniform for the best interpretation. Morecomplex costs functions may include 5D interpolation (x, y, z, time,distribution), or optimize for better offset vector tile (distributionsin the offset vector space per bin per fold tile), or more forinterpretation, i.e., optimize for a near trace gradient stack (used forAVO and rock property detection) where each bin could have at least 1hit per every 200 m offset from 0 to 800 m offset. These nonlimitingexamples of costs functions may be either dictated by acquisition theory(fold, offsets and azimuths) processing (5D interpolation, offset vectortile distribution) or interpretation (gradient stack for AVO).

A convenient choice for constructing a cost function is to use areconstruction algorithm that can produce uniformly sampled data from aset of non-uniform sample locations. Example algorithms range fromsimple linear or nearest neighbour interpolation to more sophisticatedreconstruction techniques such as MWNI (Liu and Sacchi, 2004), andcompressive sensing based reconstruction techniques (Hennefent andHerrmann, 2008, Herrmann, 2010). The cost function can be derivedindependently of the data by matrix analysis, or it can be a combinedwith prior knowledge of the underlying earth model if available.

In one nonlimiting embodiment, a compressive sensing algorithm for datareconstruction (e.g., Herrmann, 2010) may be used for calculation of thecost function. This algorithm uses a sampling matrix that extracts asubset of the data from an underlying uniformly sampled grid. Acompressive sensing algorithm is then used to reconstruct the data onthe underlying grid. The signal-to-noise ratio of the reconstructed datais used as the cost function for the outer NUOS optimization loop. Thesignal-to-noise ratio may be approximated by constructing elasticsynthetic seismograms from a detailed model of the study area. Syntheticrecords may be calculated for very dense spatial sampling, and thendecimated according to a particular realization of the non-uniformsampling matrix. The signal-to-noise ratio for a particular realizationmay be defined as the root-mean-square of the reconstructed data minusthe original synthetic data over a window corresponding to an area ofinterest.

Selection of an optimal design based on evaluations of the cost functionand associated constraints can be cast as a classic optimizationproblem, for which a wide range of potential solutions is available. AMonte-Carlo optimization scheme may be used to select the sourcelocations for the field trial.

As an example of contrasting a convention acquisition design with a NUOSdesign, a conventional uniform survey may be acquired with a fixedspread of 580 receivers spaced at 25 meters, and with a source spacingof 25 meters over the same aperture as the receiver spread. A normalmoveout processing application with a velocity function designed forminimizing aliasing between near and far offsets applied to the datawill likely nevertheless produce the result that significant aliasingwill occur between 30 and 60 Hz.

In contrast, a NUOS design may use an underlying sampling matrix with6.25 meter spacing and the number of shots identical to that used in theuniform design, and covering the same spatial aperture. Several hundredMonte Carlo iterations may be used to select optimal source locationsfor these parameters. The selection criteria used may be based on thesignal-to-noise ratio of data reconstructed from elastic wave syntheticseismograms using a detailed velocity model from the study area.Reconstructed receiver gathers using the NUOS criteria and compressivesensing algorithm produce seismic records largely free of aliasingartifacts.

Non-Uniform Optimal Sampling (NUOS) provides a methodology for choosingnon-uniform sensor locations for seismic survey planning. This techniqueuses compressive sensing along with optimization algorithms to identifysensor layouts that satisfy optimization constraints for a particularsurvey. Field trials conducted using NUOS concepts confirms theviability of using compressive sensing algorithms to recoversignificantly broader spatial bandwidth from non-uniform sampling thancould be obtained using uniform sampling. In the 2D field examplediscussed above, data with spatial bandwidth equivalent to 12.5 muniform sampling was obtained using the same number of samples as wouldbe used for 25 m survey, or one-half the effort of a conventional 2Dsurvey. This demonstrates that using NUOS methodology for a given numberof sources and receivers improvements over convention acquisition may beexpected that reduce the cost of survey for a fixed area, to cover alarger area with the same amount of equipment, or to increase theresolution over a given area.

One nonlimiting embodiment is shown in FIG. 15, a method for 2D seismicdata acquisition, so that simultaneously acquired deep profile data andshallow profile data may be acquired with one traverse of a survey areawith the acquisition equipment. The method comprises determiningsource-point seismic survey positions for a combined deep profileseismic data acquisition with a shallow profile seismic data acquisitionwherein the source-point positions are based on non-uniform optimalsampling 1502. A seismic dataset is obtained where the data wereacquired with a first set of air-guns optimized for a deep-data seismicprofile and acquired with a second set of air-guns optimized for ashallow-data seismic profile 1504. The seismic dataset acquired with thefirst and second set of air-guns is de-blended to obtain a deep 2Dseismic dataset and a shallow 2D seismic dataset 1506. In one embodimentthe first set of air-guns has a first encoded source signature and thesecond set of air-guns has a second encoded source signature, which mayfacilitate the de-blending process.

Another nonlimiting embodiment comprises using interpolated compressivesensing to reconstruct the acquired dataset to a nominal grid. Acompressive sensing application takes the originally acquired sparsedata from an underlying acquisition position, data which may be acquiredon a regular or irregular underlying grid, and moves reconstructed datato a nominal grid, which may be a uniformly sampled grid. After the dataare reconstructed using interpolated compressive sensing, the data maybe used to obtain shot-point or source-point gathers (or common sourcegathers), receiver gathers (or common detector gathers), common offsetgathers or common midpoint gathers. Each of these types of gathers mayhave beneficial utility in different aspects when the combineddeep-shallow simultaneously acquired dataset is deblended into a deepprofile dataset and shallow profile dataset. De-blending may beaccomplished as disclosed by Mahdad et al., 2011, which is expresslyincorporated herein by reference.

In still another nonlimiting embodiment, a non-uniform optimal samplingmethod further comprises using a Monte Carlo Optimization scheme todetermine source-point seismic survey positions. Determining thenon-uniform optimal sampling using a Monte Carlo Optimization scheme mayfurther comprise a Signal-to-Noise Ratio cost-function (SNRcost-function) defined as the root-mean-square SNR of the data to bereconstructed minus the SNR of an elastic wave synthetic dataset over anarea of interest using an appropriate velocity model. This is a way ofexploiting a priori knowledge of the underlying earth model to providefor favorable conditions for sparse recovery that may improve thewavefield sampling operator, rather than relying on randomness or someother non-model related parameter.

Additionally, using a Monte Carlo Optimization scheme for determiningthe non-uniform optimal sampling may comprise using a cost-function todetermine the optimized locations, the cost function may be diagonaldominance, a conventional array response, a condition number, eigenvaluedetermination, mutual coherence, trace fold, or azimuth distribution.

Embodiments disclosed herein may be used in conjunction with system 1600as illustrated in FIG. 16. FIG. 16 illustrates a schematic diagram of anembodiment of a system 1600 that may correspond to or may be part of acomputer and/or any other computing device, such as a workstation,server, mainframe, super computer, processing graph and/or databasewherein NUOS technology may be applied. The system 1600 includes aprocessor 1602, which may be also be referenced as a central processorunit (CPU). The processor 1602 may communicate and/or provideinstructions to other components within the system 1600, such as theinput interface 1604, output interface 1606, and/or memory 1608. In oneembodiment, the processor 1602 may include one or more multi-coreprocessors and/or memory (e.g., cache memory) that function as buffersand/or storage for data. In alternative embodiments, processor 1602 maybe part of one or more other processing components, such as applicationspecific integrated circuits (ASICs), field-programmable gate arrays(FPGAs), and/or digital signal processors (DSPs). Although FIG. 16illustrates that processor 1602 may be a single processor, it will beunderstood that processor 802 is not so limited and instead mayrepresent a plurality of processors including massively parallelimplementations and processing graphs comprising mathematical operatorsconnected by data streams distributed across multiple platforms,including cloud-based resources. The processor 1602 may be configured toimplement any of the methods described herein.

FIG. 16 illustrates that memory 1608 may be operatively coupled toprocessor 1602. Memory 1608 may be a non-transitory medium configured tostore various types of data. For example, memory 1608 may include one ormore memory devices that comprise secondary storage, read-only memory(ROM), and/or random-access memory (RAM). The secondary storage istypically comprised of one or more disk drives, optical drives,solid-state drives (SSDs), and/or tape drives and is used fornon-volatile storage of data. In certain instances, the secondarystorage may be used to store overflow data if the allocated RAM is notlarge enough to hold all working data. The secondary storage may also beused to store programs that are loaded into the RAM when such programsare selected for execution. The ROM is used to store instructions andperhaps data that are read during program execution. The ROM is anon-volatile memory device that typically has a small memory capacityrelative to the larger memory capacity of the secondary storage. The RAMis used to store volatile data and perhaps to store instructions.

As shown in FIG. 16, the memory 1608 may be used to house theinstructions for carrying out various embodiments described herein. Inan embodiment, the memory 1608 may comprise a computer program module1610, which may embody a computer program product, which may be accessedand implemented by processor 1602. Computer program module 1610 may be,for example, programs to implement NUOS and compressive sensingtechnology. Alternatively, application interface 1612 may be stored andaccessed within memory by processor 1602. Specifically, the programmodule or application interface may perform signal processing and/orconditioning and applying non-uniform optimal sampling and compressivesensing as described herein.

Programming and/or loading executable instructions onto memory 1608 andprocessor 1602 in order to transform the system 1600 into a particularmachine or apparatus that operates on time series data is well known inthe art. Implementing instructions, real-time monitoring, and otherfunctions by loading executable software into a computer can beconverted to a hardware implementation by well-known design rules. Forexample, decisions between implementing a concept in software versushardware may depend on a number of design choices that include stabilityof the design and numbers of units to be produced and issues involved intranslating from the software domain to the hardware domain. Often adesign may be developed and tested in a software form and subsequentlytransformed, by well-known design rules, to an equivalent hardwareimplementation in an ASIC or application specific hardware thathardwires the instructions of the software. In the same manner as amachine controlled by a new ASIC is a particular machine or apparatus,likewise a computer that has been programmed and/or loaded withexecutable instructions may be viewed as a particular machine orapparatus.

In addition, FIG. 16 illustrates that the processor 1602 may beoperatively coupled to an input interface 1604 configured to obtain dataand output interface 1606 configured to output and/or display theresults or pass the results to other processing. The input interface1604 may be configured to obtain time series and compressive sensingdata via sensors, cables, connectors, and/or communication protocols. Inone embodiment, the input interface 1604 may be a network interface thatcomprises a plurality of ports configured to receive and/or transmitNUOS technology data via a network. In particular, the network maytransmit the data via wired links, wireless link, and/or logical links.Other examples of the input interface 1604 may be universal serial bus(USB) interfaces, CD-ROMs, DVD-ROMs. The output interface 1606 mayinclude, but is not limited to one or more connections for a graphicdisplay (e.g., monitors) and/or a printing device that produceshard-copies of the generated results.

CONCLUSIONS

The seismic data acquisition design problem was considered from thepoint of view of compressive sensing seismic data reconstruction andnon-uniform optimal sampling. In particular, mutual coherence and agreedy optimization algorithm was utilized to design an optimalacquisition grid. With the synthetic example, the signal-to-noise ratioand the mutual coherence are anti-correlated. Additionally, thesynthetic example showed that the randomized greedy algorithm gave amutual coherence that is lower than that found from a Monte Carlosimulation. Further, the signal-to-noise ratio of the reconstructionresult produced from the optimal grid found through the greedy algorithmis similar to that found from the Monte Carlo simulation, which can bepredicted from the work of Hennenfent and Herrmann (2008). Finally, thechoice of mutual coherence proxy using a real data example wasvalidated, and where a qualitative analysis of the reconstructionresults was made, comparing a low mutual coherence sampling grid and ahigh mutual coherence sampling grid of the same survey area.

Although the systems and processes described herein have been describedin detail, it should be understood that various changes, substitutions,and alterations can be made without departing from the spirit and scopeof the invention as defined by the following claims. Those skilled inthe art may be able to study the preferred embodiments and identifyother ways to practice the invention that are not exactly as describedherein. It is the intent of the inventors that variations andequivalents of the invention are within the scope of the claims whilethe description, abstract and drawings are not to be used to limit thescope of the invention. The invention is specifically intended to be asbroad as the claims below and their equivalents.

REFERENCES

All of the references cited herein are expressly incorporated byreference. The discussion of any reference is not an admission that itis prior art to the present invention, especially any reference that mayhave a publication data after the priority date of this application.Incorporated references are listed again here for convenience:

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What is claimed is:
 1. A method for 2D seismic data acquisition, themethod comprising: determining source-point seismic survey positions fora combined deep profile seismic data acquisition with a shallow profileseismic data acquisition, the source-point seismic survey positionsbased on non-uniform optimal sampling using a Monte Carlo Optimizationscheme and a cost-function, wherein the source-point survey positionsare determined irrespective of any nominal grid; obtaining a seismicdataset acquired with a first set of air-guns optimized for a deep-dataseismic profile and acquired with a second set of air-guns optimized fora shallow-data seismic profile; and de-blending the seismic dataset toobtain a deep 2D seismic dataset and a shallow 2D seismic dataset. 2.The method of claim 1, further comprising: reconstructing the seismicdataset to a nominal grid using interpolated compressive sensing.
 3. Themethod of claim 2, wherein the nominal grid is a uniformly sampled grid.4. The method of claim 2, wherein reconstructing the seismic datasetimplements an analysis-based recovery including alternative directionmethod (ADM) and a variable splitting technique.
 5. The method of claim1, further comprising: reconstructing the seismic data to obtain areceiver gather.
 6. The method of claim 1, wherein the cost-function isa Signal-to-Noise Ratio cost-function (SNR cost-function) defined as theroot-mean-square SNR of the data to be reconstructed minus the SNR of anelastic wave synthetic dataset over an area of interest using anappropriate velocity model.
 7. The method of claim 1, wherein thecost-function is selected from: diagonal dominance, a conventional arrayresponse, a condition number, eigenvalue determination, mutualcoherence, trace fold, and azimuth distribution.
 8. The method of claim1, wherein the first set of air-guns has a first encoded sourcesignature and the second set of air-guns has a second encoded sourcesignature.
 9. The method of claim 1, wherein the determining of thesource-point seismic survey positions includes determining an underlyinguniformly sampled grid.
 10. A method for seismic data acquisition, themethod comprising: determining source-point seismic survey positions fora combined deep profile seismic data acquisition with a shallow profileseismic data acquisition, the source-point seismic survey positionsbased on non-uniform optimal sampling using a Monte Carlo Optimizationscheme, the Monte Carlo Optimization scheme further comprising aSignal-to-Noise Ratio cost-function (SNR cost-function) defined as theroot-mean-square SNR of the data to be reconstructed minus the SNR of anelastic wave synthetic dataset over an area of interest using anappropriate velocity model; obtaining a seismic dataset acquired using afirst set of air-guns optimized for a deep-data seismic profile and asecond set of air-guns optimized for a shallow-data seismic profile; andde-blending the seismic dataset to obtain a deep seismic dataset and ashallow seismic dataset.
 11. The method of claim 10, wherein thesource-point survey positions are determined irrespective of any nominalgrid.
 12. The method of claim 10, further comprising: reconstructing theseismic dataset to a previously undefined nominal grid usinginterpolated compressive sensing.
 13. The method of claim 12, whereinthe previously undefined nominal grid is a uniformly sampled grid. 14.The method of claim 12, wherein reconstructing the seismic datasetimplements an analysis-based recovery including alternative directionmethod (ADM) and a variable splitting technique.
 15. The method of claim14, wherein the analysis based recovery is governed by:min_(u) ∥Su∥s.t.∥Ru−b∥ ₂≤σ, where u is reconstructed seismic data, S isan appropriately chosen dictionary, R is a restriction operator, b isobserved seismic data, and σ is noise.
 16. A method for seismic dataacquisition, the method comprising: determining source-point seismicsurvey positions for a combined deep profile seismic data acquisitionwith a shallow profile seismic data acquisition, the source-pointseismic survey positions based on non-uniform optimal sampling using aMonte Carlo Optimization scheme, the Monte Carlo Optimization schemefurther comprising a cost-function to determine optimized locations, thecost-function selected from: diagonal dominance, a conventional arrayresponse, a condition number, eigenvalue determination, mutualcoherence, trace fold, and azimuth distribution; obtaining a seismicdataset acquired with a first set of air-guns optimized for a deep-dataseismic profile and a second set of air-guns optimized for ashallow-data seismic profile; and de-blending the seismic dataset toobtain a deep seismic dataset and a shallow seismic dataset.
 17. Themethod of claim 16, wherein the source-point survey positions aredetermined irrespective of any nominal grid.
 18. The method of claim 16,further comprising: reconstructing the seismic dataset to a previouslyundefined nominal grid using interpolated compressive sensing.
 19. Themethod of claim 18, wherein the previously undefined nominal grid is auniformly sampled grid.
 20. The method of claim 19, whereinreconstructing the seismic dataset implements an analysis-based recoveryincluding alternative direction method (ADM) and a variable splittingtechnique.